Epilog III - On { Triple-Cross Symmetry }

by Frank C. Fung - 1st published in November, 2007.

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Triple-Cross Symmetry :

• We start-off again with the { Cubic Close-Packed } pattern with the { Base Atom } being surrounded by { 12 atoms } :
  • as per the diagram on-the-right , below .

  

• We then construct 6 { straight-lines } joining the { mid-points } on opposite-sides of the { cube } , to form 3 { crosses } , namely :
  • the { Orange-Cross } , the { Green-Cross } , and the { Cyan-Cross } , as shown above .

  We then combine the 3 { crosses } to form the { 6-Sticks Model } on the very-right , above .

• We note here a very specific property for this { 6-Sticks Model } :
  • each pair of { different-color lines } always intersects at an angle of [ 60 degrees ] .

• And { Tri-Symmetry } is demonstrated via the following :
  • If we exchange any 2 { crosses } , the shape of the { 6-Sticks Model } remains unchanged ;

  • If we exchange the 3 { crosses } in any rotary order , the shape of the { 6-Sticks Model } also remains unchanged .

  This then is known as { Triple-Cross Symmetry } .

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This then concludes our current investigation on { 2, 3, 4, 6, 8-Symmerties } .

go to the last Section - Epilog II - On Multiplex Symmetry

go to the Home Page for A Solid Object with 14 Faces

original dated 2007-11-21